Let [Formula: see text] be a commutative ring and nonzero text]-module. We introduce the class of pseudo-strongly (PS)-hollow submodules text]. Inspired by theory modules with secondary representations, we investigate which can written as finite sums PS-hollow submodules. In particular, provide existence uniqueness theorems for minimal strongly representations over Artinian rings.

In this paper, a Gaifman-Shapiro-style module architecture is tailored to the case of Smodels programs under the stable model semantics. The composition of Smodels program modules is suitably limited by module conditions which ensure the compatibility of the module system with stable models. Hence the semantics of an entire Smodels program depends directly on stable models assigned to its modul...

Many biomedical PhD trainees lack exposure to business principles, which limits their competitiveness and effectiveness in academic and industry careers. To fill this training gap, we developed Business and Management Principles for Scientists, a semester-long program that combined didactic exposure to business fundamentals with practical team-based projects aimed at solving real business probl...

let $r$ be a commutative ring. in this paper we assert some properties of finitely generated comultiplication modules and fitting ideals of them.

we show that every semi-artinian module which is contained in a direct sum of finitely presented modules in $si[m]$, is weakly co-semisimple if and only if it is regular in $si[m]$. as a consequence, we observe that every semi-artinian ring is regular in the sense of von neumann if and only if its simple modules are $fp$-injective.

In Arabidopsis thaliana, lateral root (LR) formation is regulated by multiple auxin/indole-3-acetic acid (Aux/IAA)-AUXIN RESPONSE FACTOR (ARF) modules: (i) the IAA28-ARFs module regulates LR founder cell specification; (ii) the SOLITARY-ROOT (SLR)/IAA14-ARF7-ARF19 module regulates nuclear migration and asymmetric cell divisions of the LR founder cells for LR initiation; and (iii) the BODENLOS/I...

let r be a ring,  be an endomorphism of r and mr be a -rigid module. amodule mr is called quasi-baer if the right annihilator of a principal submodule of r isgenerated by an idempotent. it is shown that an r-module mr is a quasi-baer module if andonly if m[[x]] is a quasi-baer module over the skew power series ring r[[x; ]].

‎A module $M$ is called FI-extending if every fully invariant submodule of $M$ is essential in a direct summand of $M$‎. ‎It is not known whether a direct summand of an FI-extending module is also FI-extending‎. ‎In this study‎, ‎it is given some answers to the question that under what conditions a direct summand of an FI-extending module is an FI-extending module?

we call a module  essentially retractable if homr for all essential submodules n of m. for a right fbn ring r, it is shown that: (i)  a non-zero module  is retractable (in the sense that homr for all non-zero ) if and only if certain factor modules of m are essentially retractable nonsingular modules over r modulo their annihilators. (ii)  a non-zero module  is essentially retractable if and on...

‎we call a ring $r$ right generalized semiperfect if every simple right $r$-module is an epimorphic image of a flat right $r$-module with small kernel‎, ‎that is‎, ‎every simple right $r$-module has a flat $b$-cover‎. ‎we give some properties of such rings along with examples‎. ‎we introduce flat strong covers as flat covers which are also flat $b$-covers and give characterizations of $a$-perfe...